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Random Variables and their Properties. Learn fundamental theoretical concepts of random variables and probability distributions Describe the representation of randomness and data variability using probability distributions.
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Random Variables and their Properties • Learn fundamental theoretical concepts of random variables and probability distributions • Describe the representation of randomness and data variability using probability distributions. • Characterize probability distributions through attributes such as moments (mean, variance, skewness…) and other measures of central tendency. • Estimate parameters of probability distributions from sample data
Text Coverage (Kottegoda and Rosso) • 3.1 Random Variables, probability distributions and density functions (p86-94) • 3.2 Descriptors of Random Variables • 3.2.1 Expectation and other probability measures (p94-103) • 3.2.3 Estimation of Parameters by method of moments (p107-109), Maximum Likelihood (p112-114), Jackknife and Bootstrap (p116-117)
A random variable X is a variable whose outcomes (values) are governed by the laws of chance. Discrete Values Probability Mass Function PX(x)=Pr(X=x)
Cumulative Distribution Function FX(x)=Pr[X ≤ x]
Continuous Variable Probability density function
= 8621 mgal S = 8194 mgal n = 31 Estimation of parameters Hillsborough River at Zephyr Hills, September flows mgal
=1.1 =1.3 x 10-3 Method of Moments Gamma distribution
=8.29 Method of Moments Log-Normal distribution =0.643
Likelihood ln(L)= -311 (for gamma) ln(L)= -312 (for log normal)
Generating a random variable from a given distribution F(U) F(X) • Generate U from a uniform distribution between 0 and 1 • Solve for X=F-1(U) Basis P(X<x)=P(U<F(x))=P(F-1(U)<x) U X F-1(U) is randomly distributed with CDF F(x)
Generating a Pseudo random number • There is a lot of lore about this. Refer to: Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, (1988), Numerical Recipes in C : The Art of Scientific Computing, Cambridge University Press, New York, 735 p. • Congruential method • Each r is an integer random number between 0 and m-1. by (m-1) gives a number between 0 and 1 that repeats after at most m numbers. Numerical recipes gives "good" choices for a, c and m. • R has built in functions runif to generate uniform random numbers, as well as other distributions, e.g rnorm, rgamma.
Generating a random variable from a discrete distribution in matlab F(U) F(X) Xs=[x1 x2 x3 …] % Vector of x values Px=[p1 p2 p3 …] % Vector of pmf values CDFx=cumsum(Px) % CDF vector U=rand(1) % Uniform random number 0-1 X = Xs(find(CDFx>U,1) ) % Inverse of CDF U X